# Sum of Reciprocals of Powers as Euler Product/Proof 2

< Sum of Reciprocals of Powers as Euler Product(Redirected from Euler Product for Riemann Zeta Function/Proof 2)

## Theorem

Let $\zeta$ be the Riemann zeta function.

Let $s\in \C$ be a complex number with real part $\sigma>1$.

Then $\zeta(s) = \displaystyle\prod_p\frac1{1-p^{-s}}$
where the infinite product runs over the prime numbers.

## Proof

From Sum of Geometric Progression:

- $\dfrac 1 {1 - p^{-z} } = 1 + \dfrac 1 {p^z} + \dfrac 1 {p^{2 z} } + \cdots$

From Sum of Reciprocals of Powers is Absolutely Convergent iff Modulus of Power is Greater than One:

- $\displaystyle \sum_{n \mathop \ge 1} n^{-z}$ is absolutely convergent

- $\left\lvert{z}\right\rvert \ge 1$

Thus:

\(\displaystyle \sum_p \dfrac 1 {1 - p^{-z} }\) | \(=\) | \(\displaystyle \sum_p \left({1 + \dfrac 1 {p^z} + \dfrac 1 {p^{2 z} } + \dfrac 1 {p^{3 z} } + \cdots}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({1 + \dfrac 1 {2^z} + \dfrac 1 {2^{2 z} } + \dfrac 1 {2^{3 z} } + \cdots}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \times \, \) | \(\displaystyle \left({1 + \dfrac 1 {3^z} + \dfrac 1 {3^{2 z} } + \dfrac 1 {3^{3 z} } + \cdots}\right)\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \times \, \) | \(\displaystyle \left({1 + \dfrac 1 {5^z} + \dfrac 1 {5^{2 z} } + \dfrac 1 {5^{3 z} } + \cdots}\right)\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \times \, \) | \(\displaystyle \cdots\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 + \dfrac 1 {2^z} + \dfrac 1 {3^z} + \dfrac 1 {2^{2 z} } + \dfrac 1 {5^z}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \dfrac 1 {2^z 3^z} + \dfrac 1 {7^z} + \dfrac 1 {2^{3 z} } + \dfrac 1 {3^{2 z} }\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle + \, \) | \(\displaystyle \cdots\) | $\quad$ | $\quad$ |

The result follows from the Fundamental Theorem of Arithmetic.

$\blacksquare$

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.19$: The Series $\sum 1/ p_n$ of the Reciprocals of the Primes